The thing that I, as a tourist, really wanted to be sure I spent time with was Central Park. The first time I went to NY, I didn't get time there at all, so when I went back, I made a point to go. I honestly wish I would have scheduled more than one day there-- there are so many cool things to explore in Central Park, and the best part is walking through the park is free (though, you might want to take some cash for the buskers-- some of them are REALLY awesome).
I did the Statue of Liberty and Ellis Island both times I went to New York City. I LOVE Ellis Island and reading all of the history there-- I'd like to go back-- but if I could go back and re-do my trips, I'd skip the Statue and just go to Ellis Island. The statue is basically a long ferry ride to a long line-- not worth the time, in my opinion.
If you want to see a show, ask about tickets to a Broadway open dress rehearsal. If you can get the tickets, it's cheaper than seeing the show normally, and usually it is the EXACT same show (they go through the whole thing just like normal, but they're more likely to forget a line in a rehearsal-- when I've gone, I've usually seen ZERO difference, and it's a cheaper ticket).
A lot of it depends on your interests. If you're a history buff, I'd stop by things like Ellis Island. If you're a shopper, I think M&M world is really cool, and otherwise, there are all of the famous shops in NY that everyone tries to go to when they're in town. If you're really interested in theatre/the arts, you'll be more interested in museums, Broadway, etc...
Oh, and for food, being a tourist and not a native, definitely stop in someplace that looks like a hole in the wall for NY style pizza, and make sure you grab a bagel from a decent deli. Yum.